Show that $\lim_{t -> \infty} V(t)$ exists and is finite for non-increasing function $V(t)$, $V(t) \geq 0$ for all $t \geq 0$

Question

Suppose that $V:[0, \infty) \rightarrow \mathbb{R}$ is non-increasing and that $V(t) \geq 0$ for all $t \geq 0$.

Show that $\lim_{t -> \infty} V(t)$ exists and is finite.

Not sure how to approach this question. If someone can point me in a direction (stuff to read, videos or similar), I would be very grateful. The questions is from a course where I'm lacking some prerequisite knowledge.

I've done some reading on infimum/supremum and lower/upper bounds.

Edit:

I guess since $V(t)$ is non-increasing, $V(t_n) \leq V(t_m)$, where $t_n > t_m$.

As $t \rightarrow \infty$, it makes sense that $V(t)$ must reach the region of its greatest lower bound, the infimum. I assume a more mathematical formulation is required. I'm also a bit confused about the information given. What does the fact that $V(t) \geq 0$ for all $t \geq 0$ suggest?

Answer 1

Hint

A bounded below real set has a lower bound. Prove that $\lim\limits_{t \to \infty} V(t) =\inf \{V(t) \mid t \ge 0\}$: the lower bound is the limit.